Topology, Geometry and Physics

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)提供资金的。陶布斯可预见的未来研究将集中在微分拓扑学问题和数学物理上。下面列出了六个具体的集中主题：·使用Seiberg-Witten方程来研究三维及更高维度流形上的矢量场的动力学。·利用辛几何定义四维流形的不变量。·研究三维流形上Dirac算符频谱的几何方面·分析其他Seiberg-Witten类方程的模空间的非紧性·研究三维流形的各种Floer同调/上同调理论之间的关系·研究从流形或流形的环空间到紧致Lie群的映射的某些量子场论。这个列表上的前四个项目主题的最终目标是开发新的工具和方法来研究低维流形的微分拓扑。低维微分拓扑问题是当前许多几何和拓扑学研究的核心问题。辛和规范理论技术为研究这类流形提供了强大的工具，但它们并不是所有的都是强大的，还需要更多的工具，特别是在无知广泛而深刻的第四维。因此，这里的重点是开发新工具并磨砺旧工具。第五个项目主题的目标是一劳永逸地解决三维和四维流形的微分拓扑不变量之间的各种猜想等价。最后一个主题的目标是建立一些严格的量子场论，物理学家用来对基本粒子相互作用进行建模，也可以用来定义共形和拓扑场论。构建数学严谨的现实量子场论版本是数学物理的关键问题之一。我们的宇宙需要三个空间坐标来描述任何给定事件的位置，然后需要第四个参数，即事件的时间。一个需要四个坐标来描述的空间被称为有四个维度。任何这样的空间看起来都很枯燥，和其他任何小尺度的空间都很相似，但大尺度的结构可能非常复杂。地球表面提供了一个二维的例子。任何给定的位置都需要指定2个参数(经度和纬度)。在小尺度上(我们行人可以看到的)，任何一个地方看起来都相对平坦(模数为一座或两座山)，与其他任何地方都很相似。然而，地球表面在大尺度上是弯曲的，因为它毕竟是一个球体。一个基本的问题是确定我们宇宙的大尺度四维形状。任何这样的决心最终都将来自天文学。即便如此，确定可能的4维形状的合理列表也是一个数学问题。这项研究项目的大部分目的是开发数学工具，使我们能够确定这样的清单。事实证明，对于维度小于4的空间，列表是已知的，也有工具可以给出维度大于4的空间的适当列表。维度4似乎是最难解决的问题。这项研究的第二个重点是调查物理学家用来预测物质和能量在亚原子长度尺度上行为的某些模型方程。这些方程是量子场论的例子。现实中的量子场理论非常复杂，因此计算通常是不可能的。这里研究的模型可能会建议改进计算技术。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The foreeable future research of C. H. Taubes will concentrate on differential topology questions and on also mathematical physics. What follows lists six specific topics of concentration:· Using the Seiberg-Witten equations to study the dynamics of vector fields on manifolds of dimension three and higher. · Using symplectic geometry to define invariants of four dimensional manifolds. · Studying geometric aspects of the spectrum of the Dirac operator on 3-dimensional manifolds· Analyzing the non-compactness of moduli spaces of other Seiberg-Witten like equations.· Studying the relations between various Floer homology/cohomology theories for 3-dimensional manifolds.· Studying certain quantum field theories of maps from a manifold or the loop space of a manifold to a compact Lie group.The ultimate goal for the first four project topics on this list is to develop new tools and methods to study the differential topology of low dimensional manifolds. Questions in low dimensional differential topology are central to much of current research in both geometry and topology. Symplectic and gauge theoretic techniques provide powerful tools for studying such manifolds, but they are not known to be all powerful, More is needed, especially in dimension four where ignorance is broad and deep. Thus the focus here to develop new tools and sharpen the old ones. The goal for the fifth project topic is to settle once and for all various conjectured equivalences between differential topology invariants of three and four dimensional manifolds. The goal of the final topic is to make some rigorous quantum field theories of the sorts that are used by physicists to model elementary particle interactions, and of the sort used to define conformal and topological field theories. The construction of mathematically rigorous versions of realistic quantum field theories is one of the key concerns of mathematical physics. Our universe requires three spatial coordinates to describe the position of any given event, and then a fourth parameter, which is the time of the event. A space that requires four coordinates to describe is said to have four dimensions. Any such space looks rather boring, and much like any other at small scales, but the large scale structure can be very complicated. The surface of the earth provides a 2-dimensional example. Any given place requires 2 parameters to specify (longitude and latitude). At small scales (those visible to us pedestrians), any one place looks relatively flat (modulo a hill or two) and much like any other. However, the surface of the earth at large scales is curved as it is, after all, a sphere. A fundamental question is to determine the large scale 4-dimensional shape of our universe. Any such determination will ultimately come from Astronomy. Even so, it is a mathematical problem to determine a reasonable list of the possible 4-dimensional shapes. Most of this research project is directed towards developing the mathematical tools that will allow us to determine such a list. As it turns out, lists are known for the spaces of dimension less than 4, and tools are available that give appropriate lists for spaces of dimension greater than 4. Dimension 4 seems the toughest nut to crack. A secondary focus of this research is to investigate certain model equations that physicists use to predict the behavior of matter and energy at sub-atomic length scales. These equations are examples of quantum field theories. Realistic quantum field theories are very complicated and so calculations are often impossible to do. The models studied here may suggest improved calculational techniques.